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The Longest Perpetual Reductions in Orthogonal Expression Reduction Systems
 In: Proc. of the 3 rd International Conference on Logical Foundations of Computer Science, LFCS'94, A. Nerode and Yu.V. Matiyasevich, eds., Springer LNCS
, 1994
"... We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normaliza ..."
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We consider reductions in Orthogonal Expression Reduction Systems (OERS), that is, Orthogonal Term Rewriting Systems with bound variables and substitutions, as in the calculus. We design a strategy that for any given term t constructs a longest reduction starting from t if t is strongly normalizable, and constructs an infinite reduction otherwise. The Conservation Theorem for OERSs follows easily from the properties of the strategy. We develop a method for computing the length of a longest reduction starting from a strongly normalizable term. We study properties of pure substitutions and several kinds of similarity of redexes. We apply these results to construct an algorithm for computing lengths of longest reductions in strongly persistent OERSs that does not require actual transformation of the input term. As a corollary, we have an algorithm for computing lengths of longest developments in OERSs. 1 Introduction A strategy is perpetual if, given a term t, it constructs an infinit...
Perpetuality and Uniform Normalization in Orthogonal Rewrite Systems
 INFORMATION AND COMPUTATION
"... We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due ..."
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Cited by 7 (2 self)
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We present two characterizations of perpetual redexes, which are redexes whose contractions retain the possibility of infinite reductions. These characterizations generalize and strengthen existing criteria for the perpetuality of redexes in orthogonal Term Rewriting Systems and the calculus due to Bergstra and Klop, and others. To unify our results with those in the literature, we introduce Contextsensitive Conditional Expression Reduction Systems (CCERSs) and prove confluence for orthogonal CCERSs. We then define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. Levy's permutation ordering on reductions) infinite reductions in orthogonal CCERSs. We then prove (1) perpetuality (in a specific context) of a redex whose contraction does not erase potentially infinite arguments, which are possibly finite (i.e., strongly normalizable) arguments that may become infinite after a number of outside steps, and (2) perpetuality (in every con...
Perpetuality and Uniform Normalization
 In Proc. of the 6 th International Conference on Algebraic and Logic Programming, ALP'97
, 1997
"... . We define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i ..."
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Cited by 4 (2 self)
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. We define a perpetual onestep reduction strategy which enables one to construct minimal (w.r.t. L'evy's ordering \Theta on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted calculi, which cannot be derived from previously known perpetuality criteria. 1 Introduction The objective of this paper is to study sufficient conditions for uniform normalization, UN, of a term in an orthogonal (first or higherorder) rewrite system, and for the UN property of the rewrite system itself. Here a term is UN if ei...
Oostrom, Uniform normalisation beyond orthogonality
 Proceedings of the Twelfth International Conference on Rewriting Techniques and Applications (RTA ’01), Lecture Notes in Computer Science (2001
, 2001
"... Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal ..."
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Abstract. A rewrite system is called uniformly normalising if all its steps are perpetual, i.e. are such that if s → t and s has an infinite reduction, then t has one too. For such systems termination (SN) is equivalent to normalisation (WN). A wellknown fact is uniform normalisation of orthogonal nonerasing term rewrite systems, e.g. the λIcalculus. In the present paper both restrictions are analysed. Orthogonality is seen to pertain to the linear part and nonerasingness to the nonlinear part of rewrite steps. Based on this analysis, a modular proof method for uniform normalisation is presented which allows to go beyond orthogonality. The method is shown applicable to biclosed first and secondorder term rewrite systems as well as to a λcalculus with explicit substitutions. 1
Conservation and Uniform Normalization in Lambda Calculi With Erasing Reductions
, 2002
"... For a notion of reduction in a #calculus one can ask whether a term satises conservation and uniform normalization. Conservation means that singlestep reductions of the term preserve innite reduction paths from the term. Uniform normalization means that either the term will have no reduction path ..."
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Cited by 1 (0 self)
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For a notion of reduction in a #calculus one can ask whether a term satises conservation and uniform normalization. Conservation means that singlestep reductions of the term preserve innite reduction paths from the term. Uniform normalization means that either the term will have no reduction paths leading to a normal form, or all reduction paths will lead to a normal form.
ON HENK BARENDREGT’S FAVORITE OPEN PROBLEM
"... Abstract. H is the λtheory extending βconversion by identifying all closed unsolvables. A longstanding open problem of H. Barendregt states that the range property holds in H. Here we discuss what we know about the problem. We also make some remarks on the λtheoryHω (the closure ofHunder the ωr ..."
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Abstract. H is the λtheory extending βconversion by identifying all closed unsolvables. A longstanding open problem of H. Barendregt states that the range property holds in H. Here we discuss what we know about the problem. We also make some remarks on the λtheoryHω (the closure ofHunder the ωrule and βconversion). 1.
ABSTRACT INFORMATION PROCESSING BY AMBIGUOUS FORMAL ENTITIES
"... Formal entities like combinators, which can ambiguously act as operands as well as operators, are introduced, discussed and represented by formal expression or by treelike graphs, in order to give an intuition of a number of significant facts. We try to develop a computational model different from ..."
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Formal entities like combinators, which can ambiguously act as operands as well as operators, are introduced, discussed and represented by formal expression or by treelike graphs, in order to give an intuition of a number of significant facts. We try to develop a computational model different from a Turing machine model or from actual computers [9]. They share the feature that the essential operations needed to benefit of their existence are first some kind of coding of structured data and secondly the writing down some algorithms to process the coded data [8]. All the criticism done by J. Backus [6] applies here and it motivated the development of Functional Programming, very related to Combinatory logic [7]. Theory is knowledge of the truth; a formalism is based on a set of signs or symbols (concrete tokens) whose assemblages obey very precise formation rules and whose features mimic those of more abstract concepts; constructivity, today, means computability or representability inside a computer. Once a phenomenon is theoretically well known, the same contents of information gained by an experiment, can equally be provided by the result of a computation, obtained by feeding into a computer some (simulation) program and some data (representing the initial state or environment of the experiment), and leaving the computer to run autonomously. In such a case we may say that the program is a process explication of the phenomenon. Since today a program is mainly written in a high level language, very different kinds of data or data types are available, like booleans (or truth values), integers, (approximate) real numbers and more structured data as: arrays of...,linear sequences of...trees of..., where the dots may be replaced by any one of the previous data